Polynomial Interpolation SICLAB

8/18/2019 Polynomial Interpolation SICLAB

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DATA FITTING IN SCILAB

In this tutorial the reader can learn about data fitting, interpolation andapproximation in Scilab. Interpolation is very important in industrial applicationsfor data visualization and metamodeling.

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.


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Data Fitting www.openeering.com page 2/16

Step 1: Purpose of this tutorial

Many industrial applications require the computation of a fitting function in

order to construct a model of the data.

Two main data fitting categories are available:

Interpolation which is devoted to the development of numerical

methods with the constraint that the fitting function fits exactly all

the interpolation points (measured data);

Approximation which is devoted to the development of numerical

methods where the type of function is selected and then all

parameters are obtained minimizing a certain error indicator to

obtain the best possible approximation.

The first category is useful when data does not present noise, while the

second one is used when data are affected by error and we want to

remove error and smooth our model.

Step 2: Roadmap

In the first part we present some examples of polynomial interpolation and

approximation. After we propose exercises and remarks.

Descriptions StepsInterpolation 3-16

Approximation or curve fitting 17-20

Notes 21-22

Exercise 23

Conclusion and remarks 24-25


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Step 3: Interpolation

The idea of approximation is to replace a function with a function

selected from a given class of approximation function .

Two main cases exist:

Continuous funct ion : In this case the function is known

analytically and we want to replace it with an easier function, for

example we may want to replace a complex function with a

polynomial for which integration or differentiation are easy;

Discrete funct ion : In this case only some values of the function

are known, i.e. and we want to make a mathematical

model which is close to the unknown function such that

it is possible to establish the value of outside the knownpoints.

Example of approximat ion of a cont inuous funct ion using a piecewise

approximat ion

Step 4: Main class of interpolation function

Several families of interpolation functions exist. The most common are:

Polynomia l in terpo la t ion of degree : In this case, we

approximate data with a polynomial of degree of the form

Piecewise po lynomia l : In this case the interval is subdivided intosubintervals in which we define a polynomial approximation of low

degree with or without continuity on the derivatives between each

subinterval.

Example piecewise in terpo la t ion wi th f i rs t der ivat ive cont inuous and

discont inuous connect ions


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Step 5: Test case: Runge function

The Runge function is defined as

Typically is considered in the interval [-5,5].

In our examples we consider 7 interpolation points, denoted with ,

equally distributed in the interval [-5,5].

The following code implements the Runge function and perform

visualization.

// Define Runge function deff('[y]=f(x)','y = 1 ./(1+x.^2)');

// Interpolation points xi = linspace(-5,5,7)'; yi = f(xi);

// Data xrunge = linspace(-5,5,101)'; yrunge = f(xrunge);

// Plot Runge function scf(1); clf(1);plot(xrunge,yrunge,'b-');plot(xi,yi,'or');xlabel("x");

ylabel("y");title("Runge function");

In terpo lat ion fu nct ion


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Step 6: Piecewise constant interpolation

Piecewise constant interpolation is the simplest way to interpolate data. It

consists on locating the nearest data value and assigning the same value

to the unknown point.

Piecewise con stant in terpo la t ion

Step 7: Piecewise constant interpolation in Scilab

The Scilab command used to perform piecewise interpolation is"interp1" where the third argument is "nearest". The fourth argument

specifies if an extrapolation method should be used when the evaluation

points are outside the interval of the interpolation points.

// Interpolation // Evaluation points xval = linspace(-6,6,101)';

xx_c = xval;yy_c = interp1(xi,yi,xx_c,'nearest','extrap');

// Plot scf(3);

clf(3);plot(xrunge,yrunge,'k-');plot(xx_c,yy_c,'b-');plot(xi,yi,'or');xlabel("x");ylabel("y");title("Piecewise interpolation");legend(["Runge func";"Interp.";"Interp. val"]);

Piecewise co nstant in terpo la t ion


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Step 8: Piecewise linear interpolation

Linear interpolation is a polynomial of degree 1 that connects two points,

, and the interpolant is given by

Piecewise l inear interpolation (green) and extrapolation (red)

Step 9: Linear interpolation in Scilab

The Scilab command used to perform linear interpolation is again

"interp1" but now the third argument is "linear". We can note that the

code is similar to the previous one for piecewise interpolation.

// Interpolation xx_l = xval;yy_l = interp1(xi,yi,xx_c,'linear','extrap');

// Plot scf(4);clf(4);plot(xrunge,yrunge,'k-');plot(xx_l,yy_l,'b-');plot(xi,yi,'or');xlabel("x");

ylabel("y");title("Linear interpolation");legend(["Runge func";"Interp.";"Interp. val"]); Piecewise l inear interpolatio n


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Step 10: Polynomial interpolation

Given a set of data points , where all are different, we find

the polynomial of degree which exactly passes through these points.

Polynomial interpolations may exhibits oscillatory effects at the end of the

points. This is known as Runge phenomenon . For example, the Runge

function has this phenomenon in the interval [-5,+5] while in the interval [-

1,1] this effect is not present.

Polynomia l in terpo la t ion

Step 11: Polynomial interpolation in Scilab

The Scilab command used to perform polynomial interpolation is

"polyfit" which is included in the zip file (see references). The syntaxrequires the interpolation points and the degree of the polynomial

interpolation which is equal to the number of point minus one.

Note that the command "horner" evaluates a polynomial in a given set of

data.

// Import function exec("polyfit.sci",-1);// Interpolation xx_p = xval;[Pn] = polyfit(xi, yi, length(xi)-1);yy_p = horner(Pn,xx_p);

// Plot

scf(5);clf(5);plot(xrunge,yrunge,'k-');plot(xx_p,yy_p,'b-');plot(xi,yi,'or');xlabel("x");ylabel("y");title("Polynomial interpolation");legend(["Runge func";"Interp.";"Interp. val"]);

Polynomia l in terpo la t ion


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Step 12: Cubic spline interpolation

Cubic spline interpolation uses cubic polynomials on each interval. Then

each polynomial is connected to the next imposing further continuity

equations for the first and second derivatives.

Several kinds of splines are available and depend on how degrees of

freedom are treated. The most well-known splines are: natural, periodic,

not-a-knot and clamped.

Spl ine in terpo la t ion

Step 13: Cubic spline in Scilab

The Scilab command used to perform cubic spline interpolation is

"splin". Several types of splines exist and can be specified by setting thethird argument of the function. The evaluation of the spline is done using

the command "interp" where it is possible to specify the extrapolation

strategy.

// Splines examples d = splin(xi, yi,"not_a_knot");// d = splin(xi, yi,"natural"); // d = splin(xi, yi,"periodic");

xx_s = xval;yy_s = interp(xx_s, xi, yi, d, "linear");

// Plot scf(6);

clf(6);plot(xrunge,yrunge,'k-');plot(xx_s,yy_s,'b-');plot(xi,yi,'or');xlabel("x");ylabel("y");title("Spline interpolation");legend(["Runge func";"Interp.";"Interp. val"]);

Cubic sp l ine in terpo la t ion w i th l inear extrapo la tion


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Step 14: Radial Basis Interpolation (RBF)

Radial basis function modeling consists of writing the interpolation function

as a linear combination of basis functions that depends only on the

distance of the interpolation points . This is equal to:

Using the interpolation conditions we have to solve the

following linear system

where the element .

Many radial basis functions exist, the most famous are:

Gaussian:

Multiquadratic:

Inverse multiquadratic:

Thin plate spline:

Step 15: Gaussian RBF in Scilab

In our example we use the Gaussian RBF.

// Gaussian RBF deff('[y]=rbf_gauss(r,sigma)','y = exp(-r.^2 ./(2*sigma))');

// Plot scf(7);clf(7);r = linspace(0,3);y1 = rbf_gauss(r,0.1);y2 = rbf_gauss(r,1.0);y3 = rbf_gauss(r,2.0);plot(r,y1,'k-');plot(r,y2,'b-');

plot(r,y3,'r-');xlabel("$r$");ylabel("$\phi(r)$");title("Gaussian rbf");legend(["$\sigma = 0.1$";"$\sigma = 1.0$";"$\sigma = 2.0$"]);

Gaussian RBF for d i f ferent va lue of sigma


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Step 17: Approximation or curve fitting

When data is affected by errors, polynomial interpolation cannot be

appropriate since the approximation function is constrained to be through

the interpolation points. So it makes sense to fit the data starting from agiven class of functions and minimizing the difference between the data

and the class of functions, i.e.

The "polyfit" function computes the best least square polynomial

approximation of data. If we choose in the "polyfit" function, we

approximate data with linear function of the form , i.e. we

compute the linear least squares fitting.

Schematic example of min in terpretat ion

Step 18: 1D approximation

In this example we add noise to the function and then we

make polynomial approximation of order 1 and 2.

The critical code is reported here (only case 1):

np = 100; noise = 0.7*(rand(np,1)-0.5);x = linspace(0,2,np)';yexact = x.^2 + x;ynoise = yexact + noise;

// degree 1 approximation p1 = polyfit(x, ynoise, 1);p1val = horner(p1,x);scf(10); clf(10);plot(x,yexact,'k-'); plot(x,ynoise,'b-'); plot(x,p1val,'r-');

For details download the zip file with the source codes. Compar ison of b est f i t t ing for degree 1 and 2


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Step 19: 2D approximation

In this example we want to approximate scattered data with a linear least

square fitting in two dimensions.

Given the polynomial approximation and thescattered interpolation points , the optimal computation of the

unknown parameters requires to solve the following

overdetermined linear system

The matrix is known as the Vandermonde matrix and arises in polynomial

interpolation. The i-th row of the matrix corresponds to the polynomial

evaluated at point i-th. In our case, the i-th row corresponds to:

This is done using the Singular Value Decomposition (SVD) or

equivalently using the backslash command. Here, we report only the

solution stage. The full code can be downloaded from our web page.

// Generating random points along a plane np = 30;noise = 0.5*(rand(np,1)-0.5);

// Extract data x = rand(np,1);y = rand(np,1);

znoise = -x+2*y+noise;

// Vandermonde matrix for P(x,y) = a+b*x+c*y V = [ones(np,1),x,y];// Find coefficient i.e. minimize error norm coeff = V\znoise;

Example of least square approximat ion in 2D


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Step 20: nD linear approximation

The previous example can be easily extended to an n-dimensional

problem, i.e. we search for an approximation of the form

.

Here, we report the code for the problem where some coefficients may be

equal to zero and could be deleted from the estimation problem.

The idea here implemented consists of the following strategy:

1. Perform least square approximation of data;

2. Find the zero coefficients with some pre-fixed tolerance;

3. Re-perform least square approximation of the reduce problemwhere we have deleted the columns of the Vandermonde matrix

corresponding to the zero coefficients.

The code reported on the right estimates these coefficients. The full code

can be downloaded from our website.

n = 10; // Problem dimensions neval = 100; // Number of evaluation points pcoeff = 1:n+1; // Problem coefficient ([a0, a1, ..., an]) pcoeff([2,5,8]) = 0; // Some zero coefficients

// Evaluation points [xdata,ydata] = generatedata(pcoeff, n, neval);

// Estimate coefficients pstar = estimatecoeff(xdata, ydata);

// Find "zero" coefficient (define tolerance) tol = 0.1;zeroindex = find(abs(pstar)


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Step 21: Another polyfit function

The numerical solution of the polynomial interpolation problem requires to

solve the linear system with the Vandermonde matrix. Numerically, this

problem is ill-conditioned and requires an efficient strategy for a “correct”solution. This is performed by using, for example, QR factorization.

The function “polyfit_fulldemo.sce”, provided by Konrad Km ieciak ,

and contained in the zip file, is an alternative to the polyfit function.

function [u]=polyfit(x, y, n) // Vandermonde for k=n:-1:0

if k==n then w=0;end w=w+1;Xu(:,w)=[x.^k];

end

// QR [q r k]=qr(Xu,'0');s = inv(r) * (q' * y); // s = r \ (q' * y)for o=1:length(s)

u(find(k(:,o)>0))=s(o);end

endfunction

Step 22: Industrial applications of data fitting

Interpolation and approximation are two major techniques for constructingmathematical models. Mathematical models can substitute complex model

in real-case applications.

When the original model is complex, or when it requires long and costly

evaluations (for example with finite element analysis – FEA), a simplified

model of the original model is required.

The model of the model is often called metamodel (a.k.a. response

surfaces ) and the metamodeling technique is widely used in industrial

applications.

In real-case applications, when optimizing product or services, we need to

evaluate several times the model. This means that having a metamodel

that can be evaluated faster is, most of the time, the only way for finding

optimal solutions.

Metamodeling+

Response surface+

Self-Organizing Maps+

Neural Networks=

Industrial applications


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Step 23: Exercise on exponential decay

As an exercise, try to estimate the coefficient of a decay function of the

form:

using a linear least square fitting of data.

Hits: Use a logarithmic change of variable to reduce the problem in the

following form

Exponent ia l decay f i t t ing of data

Step 24: Concluding remarks and References

In this Scilab tutorial we have shown how to apply data fitting in Scilab

starting from piece-wise interpolation, presenting polynomial fitting and

cubic spline, and ending up with radial basis functions (RBF).

In the case of polynomial interpolation we used the function "polyfit"

which can be downloaded from the Scilab webpage and is included in the

provided source codes.

1. Scilab Web Page: Available: www.scilab.org.

2. Openeering: www.openeering.com.

3. Javier I. Carrero is the author of the polyfit function. The original

code is available on the Scilab.org web pages and is included in the

zip file.

http://www.scilab.org/http://www.scilab.org/http://www.scilab.org/http://www.openeering.com/http://www.openeering.com/http://www.openeering.com/http://www.openeering.com/http://www.scilab.org/


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Step 25: Software content

To report bugs or suggest improvements please contact the Openeering

team. We thank Konrad Kmieciak for reporting us a new version of the

polyfit function.

www.openeering.com

Thank you for your attention,

Manolo VenturinSilvia Poles

--------------Main directory--------------ex0.sce : Plotting of the first figureex1.sce : Solution of exercise 1interpolation.sce : Interpolation examples codespolyfit.sci : Polyfit functionpolyfit_manual.pdf : Polyfit manualpolyfit_fulldemo.sce : Another version of polyfitestimate_lincoeff.sce : Estimantion of nD linear modellicense.txt : The license file
http://www.openeering.com/http://www.openeering.com/http://www.openeering.com/

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Polynomial Interpolation SICLAB